A294512 Denominators of partial sums of the reciprocals of octagonal numbers.

1, 8, 168, 420, 5460, 14560, 276640, 3043040, 136936800, 136936800, 4245040800, 72165693600, 2670130663200, 2670130663200, 114815618517600, 1320379612952400, 9242657290666800, 3080885763555600, 280080523959600, 8122335194828400, 165154148961510800, 14533565108612950400, 973748862277067676800

Offset: 0

Author

Wolfdieter Lang, Nov 01 2017

Keywords

• nonn
• frac
• easy

Comments

The corresponding numerators are given in A250401.

The octagonal numbers are here A000567(k+1) = (k + 1)*(3*k + 1), k >= 0.

In general the partial sums V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits are of interest for series Sum_{k>=1} a(k)/k with a periodic sequence a(r + m*k) = a(r), r = 1..m, k >= 1, and Sum_{r=1..m} a(r) = 0. Such sequences were considered by Euler in his Introductio in Analysin Infinitorum (1748). See the Koecher reference. Namely, Sum_{k>=1} a(k)/k = Sum_{r=1..m-1} a(r)*v_m(r) with v_m(r) = ((m-r)/m)*lim_{n -> oo} V(m,r,n).

The general formula is m*v_m(r) = log(m) + (Pi/2)*cot(Pi*r/m) - Sum_{s=1..m-1} cos(2*Pi*r*s/m)*log(2*sin((Pi*s)/m)), r = 1..m-1. (Koecher, Satz, p. 191.)

Here the instance m = 3, r = 1 is considered with V(3,1;n) = Sum_{k=0..n} 1/((k + 1)*(3*k + 1)) and lim_{n -> oo} V(3,1;n) = (Pi/sqrt(3) + 3*log(3))/4 with its decimal expansion 1.277409057... given in A244645.

Examples

			The rationals V(3,1;n) begin: 1, 9/8, 197/168, 503/420, 6623/5460, 17813/14560, 340527/276640, 3763087/3043040, 169947523/136936800, 170436583/136936800, ...
V(3,1,10^4) = 1.2773757281147540626 (Maple 20 digits) to be compared with 1.2774090575596367312 (20 digits from A244645).
The series is V(3,1) =  1 + 1/(2*4) + 1/(3*6) + 1/(4*10) + ... .
		

References

• Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.

Crossrefs

Cf. A000567, A244645, A250400, A250401.

Programs

Mathematica

Denominator@ Accumulate@ Array[1/PolygonalNumber[8, #] &, 23] (* Michael De Vlieger, Nov 01 2017 *)

Formula

a(n) = denominator(V(3,1;n)) with V(3,1;n) = Sum_{k=0..n} 1/((k + 1)*(3*k + 1)) = (1/2)*Sum_{k=0..n} (3/(3*k + 1) - 1/(k+1)), n >= 0.

a(n) = A250400(n+1)/(n+1), n >= 0. [conjecture].